Core inference · Estimation
Margin of Error Calculator
How far might a survey result sit from the truth, by chance alone? Get the ± margin of error for a proportion or a mean, the confidence interval it implies, and the sample size you would need for a tighter margin. With the one warning every poll buries in the small print: the margin covers sampling error only.
Polls usually quote the margin at a proportion of 50% — the widest it can be — so the stated figure covers whatever the actual result turns out to be.
Result
In plain English
A sample is only a slice of the whole population, so its result will be a little off from the true value just by the luck of who got picked. The margin of error puts a number on that wobble: at 95% confidence, repeating the survey many times would land the estimate within ± the margin of the truth about 95% of the time. It shrinks as the sample grows — but only the chance error shrinks, never the bias.
- margin of error (MoE)
- The half-width of the confidence interval: z (or t) × the standard error. The result is reported as estimate ± MoE.
- standard error
- The spread of the estimate from sample to sample: √[p(1 − p) ∕ n] for a proportion, s ∕ √n for a mean.
- the √n law
- The margin falls with the square root of the sample size, so cutting it in half needs four times the data, not twice. Diminishing returns set in fast.
- the 50% worst case
- A proportion’s margin is widest at p = 50% and narrows toward 0% or 100%. Pollsters quote the 50% figure so the stated margin holds whatever the answer.
- what it ignores
- The margin is sampling error only. Bias from a skewed sampling frame, non-response, or a leading question is invisible to it and often far larger.
Frequently asked
How do you calculate the margin of error?
For a proportion, MoE = z × √[p(1 − p) ∕ n], where z is the critical value for your confidence level (1.96 at 95%), p the sample proportion and n the sample size. For a mean, MoE = t × s ∕ √n, using the t critical value with n − 1 degrees of freedom and the sample standard deviation s. The result is reported as your estimate plus-or-minus that margin.
What sample size do I need for a ±3% margin of error?
About 1,068 for a 95% confidence level, assuming the worst case of p = 50%. The formula is n = z² × p(1 − p) ∕ MoE² — so ±5% needs roughly 385, ±3% about 1,068, ±2% about 2,401 and ±1% around 9,604. Because the margin falls with the square root of n, each tightening costs disproportionately more respondents.
Why is the margin of error largest at 50%?
Because the variability of a proportion, p(1 − p), is greatest at p = 0.5 and shrinks toward 0 as p approaches 0% or 100%. A near-unanimous result (say 95% “yes”) is more precisely pinned down than a 50/50 split. Pollsters report the 50% margin as a single conservative figure that covers every possible result in the survey.
Does a small margin of error mean the poll is accurate?
No — only that the sampling error is small. The margin of error says nothing about bias: a poll of the wrong people, with heavy non-response or a loaded question, can be badly wrong while reporting a tiny margin. A huge sample can give a ±1% margin around a systematically skewed estimate. Always ask how the sample was drawn, not just how big it was.